Okay, I will post a brief description of my formal systems for
hierarchies of Tarskian truth predicates, which prove relatively
strong well-ordering statements (e.g., enough to imply Kruskal's
theorem) and are supposed to be predicative.
But before I do that, can someone *please* tell me what is wrong
with my critique of Feferman-Schutte? This is independent of
the success of my truth theories.
At least two messages have now been posted which say that my
objections are unconvincing, but don't say why. I suspect that
neither of the authors had actually read my critique. The full
version is available in my paper "Predicativity beyond Gamma_0",
posted on my web site at
http://www.math.wustl.edu/~nweaver/conceptualism.html
but here is a brief version.
THE CLAIM BEING CRITIQUED:
Gamma_0 is the smallest predicatively non-provable ordinal. That
is, every ordinal less than Gamma_0 is isomorphic to an ordering
of omega which can be predicatively proven to be a well-ordering;
this is not true of Gamma_0 or any larger ordinal.
THE ACCEPTED JUSTIFICATION FOR THE CLAIM:
If a predicativist trusts some formal system for second order
arithmetic, then he should accept not only the theorems of the
system itself but also additional statements such as the assertion
that the system is consistent. He should indeed accept a "formalized
omega-rule schema" applied to the original system. Then the original
system plus the schema constitutes a new system that he accepts, and
the process can be iterated. It can even be transfinitely iterated,
yielding a family of formal systems S_a indexed by ordinal notations.
Kriesel proposed that a predicativist should accept the system S_a
when and only when he has a prior proof that a is an ordinal notation.
Feferman proved that if S_0 is a reasonable base system, then Gamma_0
is the smallest ordinal with the property that there is no finite
sequence of ordinal notations a_1, ..., a_n with a_1 a notation for
0, a_n a notation for Gamma_0, and such that S_{a_i} proves that
a_{i+1} is an ordinal notation. Thus, Gamma_0 is the smallest
predicatively non-provable ordinal.
FIRST OBJECTION:
The plausibility of Kriesel's proposal hinges on our conflating two
versions of the concept "ordinal notation" --- supports transfinite
induction for arbitrary sets versus supports transfinite induction
for arbitrary properties --- which are not predicatively equivalent.
When we prove a_i is an ordinal notation in Feferman's set-up, we
are only showing transfinite induction up to a_i for statements of
the form "b is in X". That is, if we know that "everything less than
b is in X implies b is in X", then we can infer that everything up
to a_i is in X. To infer soundness of S_{a_i} we need transfinite
induction up to a_i for the statement "S_b is sound". That is a
genuinely stronger assertion since, for example, S_{a_i} proves the
existence of arithmetical jump hierarchies up to a_i. So we should
not be able to infer soundness of S_{a_i} from the fact that a_i is
an ordinal notation.
SECOND OBJECTION:
Let us grant that the predicativist can somehow make the disputed
inference. Then for each a he has some way to make the deduction
(*) from I(a) and Prov_{S_a}(A(n)), infer A(n)
for any formula A, where I(a) formalizes the assertion that a is an
ordinal notation (supporting transfinite induction for sets).
Shouldn't he then accept the assertion
(**) (forall a)(forall n)[I(a) and Prov_{S_a}(A_n) --> A(n)]
for any formula A?
It is easy to see that the second assertion implies I(a) where a
is a notation for Gamma_0. So we somehow have to accept every
instance of the rule (*) but not the general implication (**).
Why?
In my Gamma_0 paper I discuss three separate (indeed, contradictory)
attempts by Kreisel to justify this. All three seemed hopeless to me.
If my critique is truly fallacious, surely someone can explain
(1) why (*) is reasonable
(2) why (**) is not reasonable.
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.eduhttp://www.math.wustl.edu/~nweaver/